A particularly beutiful note reaching your ear from a stradivarius violin has a wavelength of 39.1cm. The room is slightly warm, so the speed of sound is 344m/s. If the string's linear density is 0.670g/m and the tension is 140N, how long is the vibrating section of the violin string?
First calculate the frequency of vibration of the sound wave.
f = Vsound/(wavelength) = 880 Hz
(which would be an A note one octave above the standard A)
Next calculate the wavelength of the waves traveling on the violin string (if waves were traveling; actually there is a standing wave there). You need to know the wave speed on the string, which is sqrt(T/density) = sqrt(140/.000670) = 457 m/s
The wavelength on the string is
Vwave/f = 0.519 m = 51.9 cm.
Assuming this is a fundamental mode of the string, the length of the vibrating string is half a wavelength, or 26 cm.
To find the length of the vibrating section of the violin string, we can use the formula for the speed of a wave on a string:
v = √(T/μ)
where:
v is the speed of the wave,
T is the tension in the string, and
μ is the linear density of the string.
Rearranging the formula, we have:
T = μv²
To find the length of the vibrating section, we need to calculate the linear mass density (μ) first:
μ = m/L
where:
m is the mass of the string, and
L is the length of the string.
Given the linear density μ = 0.670 g/m and the tension T = 140 N, we can rewrite the equation as:
140 N = (0.670 g/m) * v²
Converting the linear density to kg/m (1 g = 0.001 kg) and rearranging the equation, we have:
v² = 140 N / (0.670 kg/m)
v² = 208.96 m²/s²
v = √208.96 m/s = 14.45 m/s
Next, we can calculate the wavelength of the note:
wavelength = v / frequency
Given the speed of sound in the room is 344 m/s and the wavelength is 39.1 cm (0.391 m), we rearrange the equation:
0.391 m = 14.45 m/s / frequency
Solving for the frequency:
frequency = 14.45 m/s / 0.391 m
frequency = 37.0 Hz
Now, we can determine the length of the vibrating section of the string using the formula:
length = (speed of sound / 2) / frequency
length = (344 m/s / 2) / 37.0 Hz
length = 9.27 m
Therefore, the length of the vibrating section of the violin string is approximately 9.27 meters.